Picasso has been characterised by many as a misogynist, a bully who put “his” women on a pedestal only to knock them off it, a man who feared, as well as desired, the female body and who was a selfish, demanding, narcissistic husband, lover and even grandparent. You get the picture, recognise the cliche. But is any of it really true?

There is another side to Picasso, and an exhibition opening at the Scottish National Portrait Gallery offers a glimpse of it. The photographer Lee Miller had a relationsip with Picasso that was neither abusive nor carnal. In a word, they were friends. Lee Miller and Picasso documents that friendship through their mutual portraits – she took more than 1,000 photographs of him; he painted her portrait six times – and adds up to a much more gentle, sociable image of Picasso than biographers tend to create. But was Lee Miller the only woman to tame this minotaur?

By no means. Picasso did not just see women as sex objects. One of the greatest friendships of his life was with the gay American writer Gertrude Stein. She tells the story of their encounters in The Autobiography of Alice B Toklas, the memoir of Picasso’s Paris that she wrote in the voice of her lover Toklas, with “Gertrude Stein” appearing as a character in the third person.

It has always struck me as puzzling, if Picasso was such a misogynist, how he could have got on so well with this formidable intellectual and pioneer of gay culture. Stein used to give the young artist copies of American cartoon strips. Their friendship was warm and close, unlike her far more distant dealings with Matisse.

When Picasso painted Stein’s portrait in 1905-6 he made her face a stony mask to convey her extreme strength of character. It is a portrait that breaks the mould of western portraiture – and of images of women in art.

Think of it. At the time Picasso painted Stein, the Victorian age had barely ended. While late 19th-century French painting has some interesting images of women such as Manet’s portrait of his painter friend Eva Gonzalès, the rule is flouncy dresses and parasols. Ever since the Renaissance, the portrayal of women had been shaped by ideals of beauty and constrained social roles.

Picasso’s Portrait of Gertrude Stein turns all that upside down. Stein has escaped from the confining categories with which western art previously ensnared women. She is neither old nor young, sexual nor submissive – her stone face makes her something new on Earth. She is in command of her identity. She is modern and powerful, an Easter Island idol of enigmatic authority.

In Picasso’s much later portraits of Lee Miller, there is a comparable sense of mystery. Picasso sits her in a chair and tries to size her up as a person, painting with cartoonish cubist freedom. What are women? What are men? Picasso’s art suffers because we expect him to be much easier than he is. The greatest artist of the 20th century is, in reality, a painter of the mysteries of perception and being. His vision, properly understood, is the most liberating ever created in art. Oppressor? Look again.

• This article was amended on 22 May 2015 to update the credits on the photo caption.

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